CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Special Integrals - 1

If n is a p...

Question

If $n$ is a positive integer and $x < 1$ , show that
$\frac{1 - x^{n + 1}}{n + 1} < \frac{1 - x^{n}}{n}$ .

Open in App

Solution

We can write $\frac{1 - x^{n + 1}}{1 - x^{n}} = 1 + \frac{x^{n} (1 - x)}{1 - x^{n}}$

$\frac{1 - x}{1 - x^{n}} = \frac{1}{{(1 + x)}^{n} - x^{n}} = \frac{1}{1 + x + x^{2} + . . + x^{n - 1}}$

$\frac{1 - x^{n + 1}}{1 - x^{n}} = 1 + \frac{x^{n}}{1 + x + x^{2} + . . + x^{n - 1}}$

$\frac{1 - x^{n + 1}}{1 - x^{n}} = 1 + \frac{1}{\frac{1}{x^{n}} + \frac{1}{x^{n - 1}} + . . . + \frac{1}{x}}$

As $x < 1$

$\frac{1}{x} > 1$

$\frac{1}{x}, \frac{1}{x^{2}}, . ., \frac{1}{x^{n}}$ is greater than $1$ , and hence their sum is greater than $n$

$\frac{1}{x} + \frac{1}{x^{2}} + . . . + \frac{1}{x^{n}} > n$

$1 + \frac{1}{\frac{1}{x} + \frac{1}{x^{2}} + . . . + \frac{1}{x^{n}}} < 1 + \frac{1}{n}$

$\frac{1 - x^{n + 1}}{1 - x^{n}} < \frac{1 + n}{n}$

$\frac{1 - x^{n + 1}}{n + 1} < \frac{1 - x^{n}}{n}$

Suggest Corrections

thumbs-up

1

Similar questions

n

is a positive integer, then the coefficient of

x^{n}

in the expansion of

\frac{(1 + x)^{n}}{1 - x}

n

' is a positive integer and '

x

' is any non-zero number, then prove that

C_{0} + C_{1} . \frac{x}{2} + C_{2} . \frac{x^{2}}{3} + . . . . . + C_{n} . \frac{x^{n}}{n + 1} = \frac{(1 + x)^{n + 1} - 1}{(n + 1) x}

Q. Show that the middle term in the expansion of

(1 + x)^{2 n}

\frac{1.3.5..... (2 n - 1)}{n!}

2^{n} x^{n}

; where n is a positive integer.

X_{n} = {z = x + i y : {| z |}^{2} \leq \frac{1}{n}}

for all integers

n \geq 1

\infty ⋂ n = 1 X_{n}

X_{n} = {z = x + i y : {| z |}^{2} \leq \frac{1}{n}}

for all integers

n \geq 1

\cap_{n = 1}^{\infty} X_{n}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Special Integrals - 1

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Special Integrals - 1

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon